Kings in semicomplete multipartite digraphs

A digraph obtained by replacing each edge of a complete p-partite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicom-plete p-partite digraph, or just a semicomplete multipartite digraph. A semicomplete multipartite digraph with no cycle of length two is a multipartite tournament. In a digraph D, an r-king is a vertex q such that every vertex in D can be reached from q by a path of length at most r. Strengthening a theorem by K.M. Koh and B.P. Tan (Discrete Math. 147 (1995) 171{183) on the number of 4-kings in multipartite tournaments, we characterize semicomplete multipartite digraphs which have exactly k 4-kings for every k